We should find analytically the optimal $W >0$ which maximize the first equation subject to the second equation, where $F( \cdot )$ is comulative distribution function (CDF), and $L_0$ and $L_1$ are positive random variables. $\xi$, $\pi_0$, $\pi_1$ are constant. Also, $0<\pi_0, \pi_1<1$ and $\pi_0 + \pi_1 =1$. All variables are real. Further, if needed, we can assume that, for example, $L_0$ and $L_1$ may have Erlang or exponential distribution.

Please specify the set over which you are taking the maximum: over what variables, in what range? Also, do you really want a $\max$ (not likely to exist due to the strict inequalities) or just a $\sup$?
–
Pietro MajerNov 8 '10 at 20:51

Edit: I just realized, if indeed $\xi''$ is a constant as I have assumed, the inequality constraint can easily be converted into a bound.
$$
\begin{align}
\int_{0}^{W^U} 1-e^{-\mu \alpha}\,d\alpha = \xi'' - \epsilon\\
\frac{e^{-\mu W^{U}}}{\mu} + W^{U} - \frac{1}{\mu} = \xi'' - \epsilon\\
\end{align}
$$
Solve for $W^{U}$, and replace the above inequality constraints with:
$$ 0 \leq W \leq W^{U}
$$
You may be able to solve this using optimal control methods.

Thenks Gilead, but I want to solve it analytically.
–
Venous007Nov 13 '10 at 13:08

Well, like I said, if $\xi''$ is constant, you can apply optimal control methods (e.g. Pontryagin's Maximum Principle) after doing a bilinear transformation. OC methods are, in principle, analytical methods. However, bear in mind that analytical (closed-form) solutions do not always exist or are difficult to get. They usually involve the solution of some nasty BVP.
–
GileadNov 13 '10 at 21:50